Assume the domain and range of f are locally compact and hausdorff. Extend domain and range by taking their compactifications, and extend f to a function g that agrees with f, and maps ω in the domain to ω in the range. We will show that f is proper iff g is continuous.
Continuity of g means any open set containing omega has open preimage containing omega, hence compact pulls back to compact, and f is proper.
If f is proper, reverse the above to show g is continuous at ω. Combine with the continuity of f, and g is continuous everywhere.